The stochastic block model (SBM) is one of the most widely used generative models for network data. Many continuous-time dynamic network models are built upon the same assumption as the SBM: edges or events between all pairs of nodes are conditionally independent given the block or community memberships, which prevents them from reproducing higher-order motifs such as triangles that are commonly observed in real networks. We propose the multivariate community Hawkes (MULCH) model, an extremely flexible community-based model for continuous-time networks that introduces dependence between node pairs using structured multivariate Hawkes processes. We fit the model using a spectral clustering and likelihood-based local refinement procedure. We find that our proposed MULCH model is far more accurate than existing models both for predictive and generative tasks.